Integral Inequality for function in $W^l_2$ space

Question

For a function $u(x_1,x_2)$ of compact support in $E_2$, the following inequality holds : $$ \iint_{-\infty}^{\infty} u^4 dx_1 dx_2 \leq \iint_{-\infty}^{\infty} u^2 dx_1 dx_2 \iint_{-\infty}^{\infty} \nabla^2u dx_1 dx_2$$ I read this in Fluids by Ladyzhenskaya, but I couldn't follow the proof. Can someone give a simple proof for this or provide the name of the above lemma so that I could search for the proof.

Answer 1

This is a variant of an interpolation inequality, often referred to as Ladyzhenskaya's inequality, which is a special case of the Gagliardo-Nirenberg interpolation inequality.

A simple proof can be done in the three-dimensional case, as by Sobolev's inequality $W^{1,2} \hookrightarrow L^6$ and hence by Hölder's inequality

$$\|u\|_{L^4}^4=\int |u| |u|^3 \leq \|u\|_{L^2} \|u\|_{L^6}^3 \leq C\|u\|_{L^2} \|u\|_{W^{1,2}}^3. $$

You should be able to find a proof in any book that investigates the Navier-Stokes equations as Ladyzhenskaya's inequality is essentially in proving the existence of weak solution. For example you can consult

• 'Topics in Nonlinear Analysis' by Luc Tartar, Lemma 2 on page 35.